Final Report Summary - IEET2 (Evolution equations for composite operators and AdS/CFT integrability)
The main aim of the project was to make a link between the knowledge and the methods, available from the study of the realistic quantum field theories (such as Qunatum Chromodynamics and Standard Model of the fundamental interactions) and a mathematical physics on the one side and a wide audience on the other side.
From the scientific side the extremely high precision computations were performed and the interdisciplinary methods were proposed and developed.
The excellent playground for the study of the general properties of the quantum field theories is a maximally extended supersymmetric gauge theory, well-known due to a discovery of the famous AdS/CFT-correspondence between this theory and a theory of superstrings in a curved space. A gauge/string duality allows to extend the methods and the results which usually are used inside the theory to the dual theory. Moreover, this unified studies gave a possibility to draw a powerful mathematical method for the solution of the spectral problem in both theory. This excellent method related with the so-called exactly solvable model. The integrability already provide the exact solutions for the several problems, which were not resolved early with the usual methods. During the project the main efforts were directed to the extension of the applicability the new methods, their high-precision test and a collection of the new data for the further research.
The quantum field theories allow one to obtain quantitative results by means of calculations of the different physical quantities with the expansion over a small parameter in the framework of perturbative theory. The unique calculations in the higher order of perturbative theory were done with a direct diagrammatic technique as well as with a new remarkable approach based on the usage of the analytical properties of the related mathematical model. The direct diagrammatic computations gave a new information about so-called color-subleading corrections, which are related with the perturbative corrections on the dual string side. Up to this moment this region of the gauge/string duality is studied not very well and the new data, which were obtained during the project, gave a unique information about such corrections. With the soon upcoming results the information about the evolution equations in this region will be extracted and the different properties of the general result will be studied. Already obtained during the project results show, that the colour-subleading corrections look as the leading result, which give a hope for the applicability of the already developed methods for these corrections too.
The calculations in the higher order of the perturbative theory with the usage of the integrability allow to reconstruct the general results from the computed fixed values, or in simple words, find exactly a function from some of its points. This was done with the methods, which was proposed and developed by Dr. Velizhanin during the project. For this purpose he used a number theory, which allows to solve a system of equations when the number of the variables is much more with compare to the number of the available equations. This method gave a simple tool for the solution of the problem, which could not be solved with the conventional methods.
Studying the analytical properties of the obtained during the project results Dr. Velizhanin found the new exact corrections to the so-called double-logarithmic evolution equations both in the supersymmetric gauge theory and in the realistic Quantum Chromodynamics. This evolution equation provides the information about behaviour od the scattering of the elementary particles during their interactions in some kinematic region, which can be interesting for the studying of the internal structure of the proton and other similar subatomic particles. Surprisingly, that the method of accounting for the corrections to the evolution equation, which was found in the supersymmetric gauge theory, works very well for the Quantum Chromodynamics. This result establishes a relation between the theoretical research in the toy-models and applicability of the obtained results and methods for the realistic theory.
As in general, to obtain a high-quality product it is necessary to use and develop the new methods, as during this project the precision computations demanded the usage of the new methods, which can be used now for the solution of the other problems.
During his staying in Humboldt University of Berlin Dr. Velizhanin gave the several extended seminars for the post-docs and post-graduated students. These talks were directed to present the approaches and the methods, which are used for the studying of the realistic theories for people, who work in the field of the pure theoretical research.
Dr. Velizhanin created a number of the high-quality animations about main futures of a subatomic world and uploaded their to the public video hosting service (YouTube).
These animations are based on the talk "The structure of the proton", which was given by Dr. Velizhanin at the traditional annual "Long Night of Science" ("Lange Nacht der Wissenschaften").
This public activity can be called rather successful as the total number of views for these videos exceeds ten thousands. Some more simple videos are in preparations and will be finished very soon.
From the scientific side the extremely high precision computations were performed and the interdisciplinary methods were proposed and developed.
The excellent playground for the study of the general properties of the quantum field theories is a maximally extended supersymmetric gauge theory, well-known due to a discovery of the famous AdS/CFT-correspondence between this theory and a theory of superstrings in a curved space. A gauge/string duality allows to extend the methods and the results which usually are used inside the theory to the dual theory. Moreover, this unified studies gave a possibility to draw a powerful mathematical method for the solution of the spectral problem in both theory. This excellent method related with the so-called exactly solvable model. The integrability already provide the exact solutions for the several problems, which were not resolved early with the usual methods. During the project the main efforts were directed to the extension of the applicability the new methods, their high-precision test and a collection of the new data for the further research.
The quantum field theories allow one to obtain quantitative results by means of calculations of the different physical quantities with the expansion over a small parameter in the framework of perturbative theory. The unique calculations in the higher order of perturbative theory were done with a direct diagrammatic technique as well as with a new remarkable approach based on the usage of the analytical properties of the related mathematical model. The direct diagrammatic computations gave a new information about so-called color-subleading corrections, which are related with the perturbative corrections on the dual string side. Up to this moment this region of the gauge/string duality is studied not very well and the new data, which were obtained during the project, gave a unique information about such corrections. With the soon upcoming results the information about the evolution equations in this region will be extracted and the different properties of the general result will be studied. Already obtained during the project results show, that the colour-subleading corrections look as the leading result, which give a hope for the applicability of the already developed methods for these corrections too.
The calculations in the higher order of the perturbative theory with the usage of the integrability allow to reconstruct the general results from the computed fixed values, or in simple words, find exactly a function from some of its points. This was done with the methods, which was proposed and developed by Dr. Velizhanin during the project. For this purpose he used a number theory, which allows to solve a system of equations when the number of the variables is much more with compare to the number of the available equations. This method gave a simple tool for the solution of the problem, which could not be solved with the conventional methods.
Studying the analytical properties of the obtained during the project results Dr. Velizhanin found the new exact corrections to the so-called double-logarithmic evolution equations both in the supersymmetric gauge theory and in the realistic Quantum Chromodynamics. This evolution equation provides the information about behaviour od the scattering of the elementary particles during their interactions in some kinematic region, which can be interesting for the studying of the internal structure of the proton and other similar subatomic particles. Surprisingly, that the method of accounting for the corrections to the evolution equation, which was found in the supersymmetric gauge theory, works very well for the Quantum Chromodynamics. This result establishes a relation between the theoretical research in the toy-models and applicability of the obtained results and methods for the realistic theory.
As in general, to obtain a high-quality product it is necessary to use and develop the new methods, as during this project the precision computations demanded the usage of the new methods, which can be used now for the solution of the other problems.
During his staying in Humboldt University of Berlin Dr. Velizhanin gave the several extended seminars for the post-docs and post-graduated students. These talks were directed to present the approaches and the methods, which are used for the studying of the realistic theories for people, who work in the field of the pure theoretical research.
Dr. Velizhanin created a number of the high-quality animations about main futures of a subatomic world and uploaded their to the public video hosting service (YouTube).
These animations are based on the talk "The structure of the proton", which was given by Dr. Velizhanin at the traditional annual "Long Night of Science" ("Lange Nacht der Wissenschaften").
This public activity can be called rather successful as the total number of views for these videos exceeds ten thousands. Some more simple videos are in preparations and will be finished very soon.